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442
Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs al ..."
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Cited by 505 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
Weierstrass and Approximation Theory
"... We discuss and examine Weierstrass' main contributions to approximation theory. ..."
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Cited by 189 (7 self)
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We discuss and examine Weierstrass' main contributions to approximation theory.
Networks and the Best Approximation Property
 Biological Cybernetics
, 1989
"... Networks can be considered as approximation schemes. Multilayer networks of the backpropagation type can approximate arbitrarily well continuous functions (Cybenko, 1989# Funahashi, 1989# Stinchcombe and White, 1989). Weprovethatnetworks derived from regularization theory and including Radial Bas ..."
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Cited by 143 (8 self)
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Networks can be considered as approximation schemes. Multilayer networks of the backpropagation type can approximate arbitrarily well continuous functions (Cybenko, 1989# Funahashi, 1989# Stinchcombe and White, 1989). Weprovethatnetworks derived from regularization theory and including Radial Basis Functions (Poggio and Girosi, 1989), have a similar property.From the point of view of approximation theory, however, the property of approximating continuous functions arbitrarily well is not sufficientforcharacterizing good approximation schemes. More critical is the property of best approximation. The main result of this paper is that multilayer networks, of the type used in backpropagation, are not best approximation. For regularization networks (in particular Radial Basis Function networks) we prove existence and uniqueness of best approximation.
Learning nearoptimal policies with Bellmanresidual minimization based fitted policy iteration and a single sample path
 MACHINE LEARNING JOURNAL (2008) 71:89129
, 2008
"... ..."
Counting faces of randomlyprojected polytopes when the projection radically lowers dimension
 J. of the AMS
, 2009
"... 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications in statistics, probability, information theory, and signal processing with ..."
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Cited by 110 (5 self)
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1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications in statistics, probability, information theory, and signal processing with potential impacts in
Is Gauss Quadrature Better Than Clenshaw–Curtis?
, 2008
"... We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to exp ..."
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Cited by 109 (4 self)
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We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z +1)/(z − 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [−1, 1].
Wavelets on graphs via spectral graph theory
, 2009
"... We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. ..."
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Cited by 90 (8 self)
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We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T t g = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
Fast evaluation of radial basis functions: Methods for twodimensional polyharmonic splines
 IMA Journal of Numerical Analysis
, 1997
"... Abstract. A generalised multiquadric radial basis function is a function of the form s(x) =∑N i=1 diφ(x − ti), where φ(r) = r2 + τ2 ..."
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Cited by 90 (5 self)
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Abstract. A generalised multiquadric radial basis function is a function of the form s(x) =∑N i=1 diφ(x − ti), where φ(r) = r2 + τ2
Learning intersections and thresholds of halfspaces
, 2004
"... We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog n ..."
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Cited by 89 (33 self)
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We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomialweight halfspaces under any distribution on the Boolean hypercube. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel nongeometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct lowdegree polynomial threshold functions. Finally, we also observe that any function of a constant number of polynomialweight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.
Counting faces of randomly projected polytopes when the projection radically lowers dimension
 Journal of the American Mathematical Society
, 2009
"... 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes, a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications – in statistics, probability, information theory, and signal processing – wit ..."
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Cited by 89 (7 self)
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1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes, a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications – in statistics, probability, information theory, and signal processing – with potential